I have heard that matrilineal descent (i.e. having Judaism passed down through the mother) comes from the Roman period when women were raped by Roman soldiers and the babies were then said to be Jewish because they knew for certain who the mother was, but they did not know if the father was Jewish. Is there any truth to this claim?

You are half right -- it most likely came from the Roman period, but not because of persecution. Rather, Roman law itself followed matrilineal descent and the Jews simply adopted this Roman practice. See Shaye Cohen's "The Beginning of Jewishness" where he lists 7 possible explanations, and the Roman one is considered one of the most likely
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CuriouserApr 5 '11 at 6:18

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"Roman law followed matrilineal descent" - say what? Nomina and cognomina were passed down in the male line, as was inheritance and so forth.
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AlexApr 5 '11 at 15:39

From "The Beginning of Jewishihness" p294: "According to Roman law, a child is the legal heir of his father and is his father's custody (potestas) only if his father and mother were joined in legal marriage (justum matrimonium). The capacity to contract a legal marriage was called conubium (or ius conubii) and was possessed almost exclusively by Roman citizens. Marriage between a person with conubium and a person without conubium was valid, but it was not a justum matrimonium; and without justum matrimonium, the status of the child follows that of its mother. Consequently, if a Roman citizen..
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CuriouserApr 5 '11 at 16:30

Sounds like the Roman law and halachah are indeed different, then; in the former the child's status follows its mother except when it doesn't (the last two cases), while in halachah the rule that Jewishness follows the mother is absolute. I think it's far more likely that such similarity as exists between them is based on the biological facts - it is indeed easier, outside of a wedded relationship, to establish motherhood than fatherhood - rather than on intercultural borrowing.
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AlexApr 5 '11 at 17:10

2. And Shecaniah the son of Jehiel, one of the sons of Elam, answered and said unto Ezra: 'We have broken faith with our God, and have married foreign women of the peoples of the land; yet now there is hope for Israel concerning this thing.

3. Now therefore let us make a covenant with our God to put away all the wives, and such as are born of them, according to the counsel of the LORD, and of those that tremble at the commandment of our God; and let it be done according to the law.

Interestingly, Ezra himself only requests the banishment of the women in 4:10, whereas as you quoted Shecaniah ben Jehiel included the women and the children in 4:2-3. One might think that Shecaniah ben Jehiel was going beyond what Ezra actually required?
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CuriouserApr 5 '11 at 17:26

The other way around, actually - Shecaniah's suggestion preceded Ezra's implementation. But after all, divorce requires a judicial proceeding, while sending children away does not - so in describing what Ezra oversaw, naturally it would mention only the former.
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AlexApr 5 '11 at 18:45

In Lev. 24:10ff, we have the story of "the son of an Israelite woman, who was also the son of an Egyptian man" who blasphemed Hashem's name and was executed judicially for this offense. Now it is true that, according to halachah, non-Jews are liable to death for this too (Rambam, Laws of Kings 9:3), but the specific mode of execution that was used, stoning, is used only for Jews; a non-Jew who blasphemes is beheaded (ibid. 9:14). So we see that he was considered a Jew because his mother was one.

In I Kings 7:14, Chiram, the artisan whom King Shlomo commissioned to make the copper vessels for the Beis Hamikdash, is described as "the son of a widow from the tribe of Naftali, and his father was a Tyrian, a coppersmith."

Now, most of the commentaries, noting that in II Chron. 2:13 his mother is described as "from the daughters of Dan," reconcile this by explaining that Chiram's father was indeed a Jew from the tribe of Naftali, who resided in Tyre (i.e., "Tyrian" describes his citizenship, not his nationality), and his wife (Chiram's mother) was from Dan.

However - and this is where it becomes relevant to the question of matrilineal descent - Abarbanel (and also Malbim) follow the straightforward understanding of the verse, and explain that Chiram's father was indeed a Tyrian gentile who had married a Jewish woman (and it was her parents who were, respectively, from the tribes of Naftali and Dan). Nonetheless, we see that Chiram is identified as a Jew.

Actually, your first proof (from Lev 24:10) is well known as a proof to the opposite. The Sifra there (quoted in the Ramban) states that the "son" in question converted to Judaism. Well, why would he need to convert, unless we say the patrilineal descent were normative? The Ramban offers an answer, but it's not terribly compelling in my humble opinion, and thus there is a strong impression that the Sifra represents a dissenting view against matrilineal descent (like a few found throughout the Yerushalmi)
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CuriouserApr 5 '11 at 17:13

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Okay, at most you've got a proof that according to this Sifra, if either parent is not Jewish then the child is not (an opinion also cited in the Gemara). Where's any proof for patrilineal descent being dispositive?
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AlexApr 5 '11 at 18:47

Well it's consistent with the view expressed by Yaakov Ish Kefar Nvuraya, in the Yerushalmi (Yevamos 2:6); a view which was condemned, but obviously was a view which circulated at a certain time and could have been adopted by the authors of the Sifra. If you're looking for something dispositive, I think you will be disappointed, but there are hints and allegations.
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CuriouserApr 5 '11 at 18:55

The Mishna in Kiddushin 66b states that if a child's mother is not Jewish, then the child is "like her," (i.e., not Jewish). This Halacha is codified in the Shulchan Aruch, Even HaEzer 8:5, without mention of any dissenting opinion. No source in the Torah teaches otherwise, and this question has never been raised in any classical Halachic text. It is an obvious and accepted axiom given to us at Sinai.

"No source in the Torah teaches otherwise" That's contested. An example is that almost all the lineage passages in the Torah trace descent through the father, regardless of the heritage of the mother.
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Judah HimangoMay 11 '11 at 14:37

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@Judah - No one contests that a persons family is traced through their father (tribal affiliation and inheritance are two important areas defined by patrilineal descent). However, which passages in the Torah imply that affiliation with the Jewish people is passed through the father and not the mother?
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Yaakov EllisMay 11 '11 at 14:44

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@Judah - it may sound like a false dichotomy to you, but it is the dichotomy defined by halacha. A person's tribal affiliation is determined by their father. A person's Jewish-ness is determined by their mother. I made up the term "family descent" because it makes sense to me in the context, but the term is not the important part. Aside from your approach making sense to you and suggesting viability, do you have any sources to back yourself up with?
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Yaakov EllisMay 11 '11 at 17:46

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@Judah - No, if they are not Jewish, then they are not Jewish (though they would have been from Binyamin had their mother been Jewish). But once they are born Jewish, then their tribe comes from the father.
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Yaakov EllisMay 11 '11 at 18:51

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@Judah - If he had married her after the giving of the Torah - yes, they would have been gentile. Being that he married her before the giving of the Torah, it is not so clear as we address it (how do you define who is a member of the Jewish nation before the Jewish nation was created?)
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Yaakov EllisMay 14 '11 at 19:01

When the Lord, your God, brings you into the land to to which you are
coming to possess it, He will cast
away many nations from before you: the
Hittites, the Girgashites, the
Amorites, the Canaanites, the
Perizzites, the Hivvites, and the
Jebusites, seven nations more numerous
and powerful that you.

And the Lord, your God, will deliver them to you, and you shall
smite them. You shall utterly destroy
them; neither shall you make a
covenant with them, nor be gracious to
them.

You shall not intermarry with them; you shall not give your daughter to
his son, and you shall not take his
daughter for your son.

For he will turn away your son from following Me, and they will worship
the gods of others, and the wrath of
the Lord will be kindled against you,
and He will quickly destroy you.

If the son of the gentile will marry your daughter, he will turn away
your son (grandson) whom your daughter
will bear for him, from following
after Me. From this we learn that your
daughter's son, who is born of a
gentile, is considered your son; but
your son's son, born of a gentile
woman, is not considered your son, but
rather her son; for it is not said
[regarding the prohibition:] "Do not
take his daughter for you son,"
because she will turn [your (grand)son
away from Me], but only that he (the
gentile) will turn your son, etc.

As an aside (since it was asked above):
The Lubavitcher Rebbe once wrote in a letter (I don't have the source right now), that it makes sense logically that Jewishness goes after the mother and lineage (e.g Cohen or Levi) goes after the father:

Since the essential existence of the baby is developed entirely by the mother in her womb, it makes sense that the essential nature of the baby (i.e. jewish or not) is determined by the nature of the mother.

On the other hand, being a Cohen or Levi involves practical action (namely, serving in the Holy Temple), and it is the father who is obligated to teach the son. Also, the father is the one who is obligated and actually performs the service, not the mother. So the child learns from the father how to be a Cohen or Levi. Therefore it makes sense that lineage goes after the father.